Electronic specific heat

When a metallic system is heated from absolute zero, not every electron gains an energy ${\displaystyle k_{\rm {B}}T}$ as equipartition dictates. Only those electrons in atomic orbitals within an energy range of ${\displaystyle {\tfrac {3}{2}}k_{\rm {B}}T}$ of the Fermi level are thermally excited. Electrons, in contrast to a classical gas, can only move into free states in their energetic neighbourhood. The one-electron energy levels are specified by the wave vector ${\displaystyle k}$ through the relation ${\displaystyle \epsilon (k)=\hbar ^{2}k^{2}/2m}$ with ${\displaystyle m}$ the electron mass. This relation separates the occupied energy states from the unoccupied ones and corresponds to the spherical surface in k-space. As ${\displaystyle T\rightarrow 0}$ the ground state distribution becomes:

${\displaystyle f={\begin{cases}1&{\mbox{if }}\epsilon _{f}<\mu ,\\0&{\mbox{if }}\epsilon _{f}>\mu .\\\end{cases}}}$

where

• ${\displaystyle f}$ is the Fermi–Dirac distribution
• ${\displaystyle \epsilon _{f}}$ is the energy of the energy level corresponding to the ground state
• ${\displaystyle \mu }$ is the ground state energy in the limit ${\displaystyle T\rightarrow 0}$, which thus still deviates from the true ground state energy.

This implies that the ground state is the only occupied state for electrons in the limit ${\displaystyle T\rightarrow 0}$, the ${\displaystyle f=1}$ takes the Pauli exclusion principle into account. The internal energy ${\displaystyle U}$ of a system within the free electron model is given by the sum over one-electron levels times the mean number of electrons in that level:

${\displaystyle U=2\sum _{k}\epsilon (\mathbf {k} )f(\epsilon (\mathbf {k} ))}$

where the factor of 2 accounts for the spin up and spin down states of the electron.

Using the approximation that for a sum over a smooth function ${\displaystyle F(k)}$ over all allowed values of ${\displaystyle k}$ for finite large system is given by:

${\displaystyle F(\mathbf {k} )={\frac {V}{8\pi ^{3}}}\sum _{k}F(\mathbf {k} )\Delta \mathbf {k} }$

where ${\displaystyle V}$ is the volume of the system.

For the reduced internal energy ${\displaystyle u=U/V}$ the expression for ${\displaystyle U}$ can be rewritten as:

${\displaystyle u=\int {\frac {d\mathbf {k} }{4\pi ^{3}}}\epsilon (\mathbf {k} )f(\epsilon (\mathbf {k} ))}$

and the expression for the electron density ${\displaystyle n={\frac {N}{V}}}$ can be written as:

${\displaystyle n=\int {\frac {d\mathbf {k} }{4\pi ^{3}}}f(\epsilon (\mathbf {k} ))}$

The integrals above can be evaluated using the fact that the dependence of the integrals on ${\displaystyle \mathbf {k} }$ can be changed to dependence on ${\displaystyle \epsilon }$ through the relation for the electronic energy when described as free particles, ${\displaystyle \epsilon (k)=\hbar ^{2}k^{2}/2m}$, which yields for an arbitrary function ${\displaystyle G}$:

${\displaystyle \int {\frac {d\mathbf {k} }{4\pi ^{3}}}G(\epsilon (\mathbf {k} ))=\int _{0}^{\infty }{\frac {k^{2}dk}{\pi ^{2}}}G(\epsilon (\mathbf {k} ))=\int _{-\infty }^{\infty }d\epsilon D(\epsilon )G(\epsilon )}$

with ${\displaystyle D(\epsilon )={\begin{cases}{\frac {m}{\hbar ^{2}\pi ^{2}}}{\sqrt {\frac {2m\epsilon }{\hbar ^{2}}}}&{\mbox{if }}\epsilon >0,\\0&{\mbox{if }}\epsilon <0\\\end{cases}}}$ which is known as the density of levels or density of states per unit volume such that ${\displaystyle D(\epsilon )d\epsilon }$ is the total number of states between ${\displaystyle \epsilon }$ and ${\displaystyle \epsilon +d\epsilon }$ . Using the expressions above the integrals can be rewritten as:

${\displaystyle {\begin{aligned}u&=\int _{-\infty }^{\infty }d\epsilon D(\epsilon )\epsilon f(\epsilon )\\n&=\int _{-\infty }^{\infty }d\epsilon D(\epsilon )f(\epsilon )\end{aligned}}}$

These integrals can be evaluated for temperatures that are small compared to the Fermi temperature by applying the Sommerfeld expansion and using the approximation that ${\displaystyle \mu }$ differs from ${\displaystyle \epsilon _{f}}$ for ${\displaystyle T=0}$ by terms of order ${\displaystyle T^{2}}$. The expressions become:

${\displaystyle {\begin{aligned}u&=\int _{0}^{\epsilon _{f}}\epsilon D(\epsilon )d\epsilon +\epsilon _{f}\left((\mu -\epsilon _{f})D(\epsilon _{f})+{\frac {\pi ^{2}}{6}}(k_{\rm {B}}T)^{2}{\dot {D}}(\epsilon _{f})\right)+{\frac {\pi ^{2}}{6}}(k_{\rm {B}}T)^{2}D(\epsilon _{f})+{\mathcal {O}}(T^{4})\\n&=\int _{0}^{\epsilon _{f}}D(\epsilon )d\epsilon +\left((\mu -\epsilon _{f})D(\epsilon _{f})+{\frac {\pi ^{2}}{6}}(k_{\rm {B}}T)^{2}{\dot {D}}(\epsilon _{f})\right)\end{aligned}}}$

For the ground state configuration the first terms (the integrals) of the expressions above yield the internal energy and electron density of the ground state. The expression for the electron density reduces to ${\displaystyle (\mu -\epsilon _{f})D(\epsilon _{f})+{\frac {\pi ^{2}}{6}}(k_{\rm {B}}T)^{2}{\dot {D}}(\epsilon _{f})=0}$. Substituting this into the expression for the internal energy, one finds the following expression:

${\displaystyle u=u_{0}+{\frac {\pi ^{2}}{6}}(k_{\rm {B}}T)^{2}D(\epsilon _{f})}$

The contributions of electrons within the free electron model is given by:

${\displaystyle C_{v}=\left({\frac {\partial u}{\partial T}}\right)_{n}={\frac {\pi ^{2}}{3}}k_{\rm {B}}^{2}TD(\epsilon _{f})}$, for free electrons : ${\displaystyle C_{V}=C_{v}/n={\frac {\pi ^{2}}{2}}{\frac {k_{\rm {B}}^{2}T}{\epsilon _{f}}}}$

Compared to the classical result (${\displaystyle C_{V}={\tfrac {3}{2}}k_{\rm {B}}}$), it can be concluded that this result is depressed by a factor of ${\displaystyle {\frac {\pi ^{2}}{3}}{\frac {k_{\rm {B}}T}{\epsilon _{f}}}}$ which is at room temperature of order of magnitude ${\displaystyle 10^{-2}}$. This explains the absence of an electronic contribution to the heat capacity as measured experimentally.

Note that in this derivation ${\displaystyle \epsilon _{f}}$ is often denoted by ${\displaystyle E_{\rm {F}}}$ which is known as the Fermi energy. In this notation, the electron heat capacity becomes:

${\displaystyle C_{v}={\frac {\pi ^{2}}{3}}k_{\rm {B}}^{2}TD(E_{\rm {F}})}$ and for free electrons : ${\displaystyle C_{V}={\frac {\pi ^{2}}{2}}k_{\rm {B}}\left({\frac {k_{\rm {B}}T}{E_{\rm {F}}}}\right)={\frac {\pi ^{2}}{2}}k_{\rm {B}}\left({\frac {T}{T_{\rm {F}}}}\right)}$ using the definition for the Fermi energy with ${\displaystyle T_{\rm {F}}}$ the Fermi temperature.

The calculation of the electron heat capacity for super conductors can be done in the BCS theory. The entropy of a system of fermionic quasiparticles, in this case Cooper pairs, is:

${\displaystyle S(T)=-2k_{\rm {B}}\sum _{k}[f_{k}\ln f_{k}+(1-f_{k})\ln(1-f_{k})]}$

where ${\displaystyle f_{k}}$ is the Fermi–Dirac distribution ${\displaystyle f_{k}={\frac {1}{e^{\beta \omega _{k}}+1}}}$ with ${\displaystyle \omega _{k}={\sqrt {\epsilon _{k}^{2}+\Delta _{k}(T)^{2}}}}$ and

• ${\displaystyle \epsilon _{k}=E_{K}-\mu =\hbar ^{2}\mathbf {k} ^{2}/2m-\mu }$ is the particle energy with respect to the Fermi energy
• ${\displaystyle \Delta _{k}(T)=-\sum _{kk'}u_{k'}v_{k'}}$ the energy gap parameter where ${\displaystyle u_{k}}$ and ${\displaystyle v_{k}}$ represents the probability that a Cooper pair is occupied or unoccupied respectively.

The heat capacity is given by ${\displaystyle C_{v}(T)=T{\frac {\partial S(T)}{\partial T}}=T\sum _{k}{\frac {\partial S}{\partial f_{k}}}{\frac {\partial f_{k}}{\partial T}}}$. The last two terms can be calculated:

${\displaystyle {\begin{aligned}{\frac {\partial S}{\partial f_{k}}}&=-2k_{\rm {B}}\ln {\frac {f_{k}}{1-f_{k}}}=2{\frac {1}{T}}{\sqrt {\epsilon _{k}^{2}+\Delta _{k}(T)^{2}}}\\{\frac {\partial f_{k}}{\partial T}}&={\frac {1}{k_{\rm {B}}T^{2}}}{\frac {e^{\beta \omega _{k}}}{(e^{\beta \omega _{k}}+1)^{2}}}\left({\sqrt {\epsilon _{k}^{2}+\Delta _{k}(T)^{2}}}-T{\frac {\partial }{\partial T}}{\sqrt {\epsilon _{k}^{2}+\Delta _{k}(T)^{2}}}\right)\end{aligned}}}$

Substituting this in the expression for the heat capacity and again applying that the sum over ${\displaystyle \mathbf {k} }$ in the reciprocal space can be replaced by an integral in ${\displaystyle \epsilon }$ multiplied by the density of states ${\displaystyle D(E_{\rm {F}})}$ this yields:

${\displaystyle C_{v}(T)={\frac {2D(E_{\rm {F}})}{k_{\rm {B}}T^{2}}}\int _{-\infty }^{\infty }\left[{\frac {e^{\frac {\sqrt {\epsilon _{k}^{2}+\Delta _{k}(T)^{2}}}{k_{\rm {B}}T}}}{(e^{\frac {\sqrt {\epsilon _{k}^{2}+\Delta _{k}(T)^{2}}}{k_{\rm {B}}T}}+1)^{2}}}\left(\epsilon _{k}^{2}+\Delta _{k}(T)^{2}-{\frac {T}{2}}{\frac {\partial }{\partial T}}\Delta _{k}(T)^{2}\right)\right]d\epsilon _{k}}$

To examine the typical behaviour of the electron heat capacity for species that can transition to the superconducting state, three regions must be defined:

1. Above the critical temperature ${\displaystyle T>T_{c}}$
2. At the critical temperature ${\displaystyle T=T_{c}}$
3. Below the critical temperature ${\displaystyle T<T_{c}}$

For ${\displaystyle T>T_{c}}$ it holds that ${\displaystyle \Delta _{k}(T)=0}$ and the electron heat capacity becomes:

${\displaystyle C_{v}(T)={\frac {4D(E_{\rm {F}})}{k_{\rm {B}}T^{2}}}\int _{-\infty }^{\infty }{\frac {e^{\beta \epsilon }}{(e^{\beta \epsilon }+1)^{2}}}\epsilon ^{2}d\epsilon ={\frac {\pi ^{2}}{3}}D(E_{\rm {F}})k_{\rm {B}}^{2}T}$

This is just the result for a normal metal derived in the section above, as expected since a superconductor behaves as a normal conductor above the critical temperature.

For ${\displaystyle T<T_{c}}$ the electron heat capacity for super conductors exhibits an exponential decay of the form: ${\displaystyle C_{v}(T)\approx e^{-\beta \Delta _{k}(0)}}$

At the critical temperature the heat capacity is discontinuous. This discontinuity in the heat capacity indicates that the transition for a material from normal conducting to superconducting is a second order phase transition.